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26. An Application of Nash-Moser Theorem to Smooth Solutions of One-Dimensional Compressible Euler Equation with Gravity

Author

Hsu, Cheng-Hsiung, Lin, Song-Sun, Makino, Tetu, and Yang, Chi-Ru

Subjects
Mathematics - Analysis of PDEs, 35L65, 35L05, 35L72
Abstract

We study one-dimensional motions of polytropic gas governed by the compressible Euler equations. The problem on the half space under a constant gravity gives an equilibrium which has free boundary touching the vacuum and the linearized approximation at this equilibrium gives time periodic solutions. But it is not easy to justify the existence of long-time true solutions for which this time periodic solution is the first approximation. The situation is in contrast to the problem of free motions without gravity. The reason is that the usual iteration method for quasilinear hyperbolic problem cannot be used because of the loss of regularities which causes from the touch with the vacuum. Interestingly, the equation can be transformed to a nonlinear wave equation on a higher dimensional space, for which the space dimension, being larger than 4, is related to the adiabatic exponent of the original one-dimensional problem. We try to find a family of solutions expanded by a small parameter. Applying the Nash-Moser theory, we justify this expansion.The application of the Nash-Moser theory is necessary for the sake of conquest of the trouble with loss of regularities, and the justification of the applicability requires a very delicate analysis of the problem.

Published
2012

32. Efficient extraction of impurities from phosphogypsum during crystal regulation of α‐hemihydrate gypsum.

Author

Guan, Qingjun, Sun, Ning, Bu, Yongjie, Fan, Ruihua, Zhang, Zhenyue, Yu, Weijian, Chi, Ru‐an, and Gao, Zhiyong

Subjects
GYPSUM, PHOSPHOGYPSUM, PHASE transitions, CRYSTAL surfaces, SOLID waste, CRYSTALS
Abstract

Phosphogypsum (PG) is a solid waste generated during the "wet phosphoric acid process." The phosphorus (P) impurities and low‐value‐added recycling products seriously constrain PG re‐utilization, which results in its massive accumulation and thus occupies large land areas and poses a severe pollution threat to the environment. In this study, by regulating gypsum crystal and P species in CaCl2–HCl solutions under mild conditions, efficient removal of P impurities (leaching efficiency of 97.78%) and synchronous preparation of high‐strength α‐CaSO4·0.5H2O (α‐HH) (compressive strength of 35.2 MPa) from PG were achieved during the phase transition from CaSO4·2H2O (DH) to α‐HH, which was a reaction of DH dissolution, followed by α‐HH crystallization. The co‐crystalline P was fully released during the dissolution process, which was necessary to efficiently eliminate P impurities. HCl dissolved the released P and transformed them into the protonated specie (H3PO4) with less similarity to SO42−, which prevented the recombination of the released P with gypsum during the crystallization process. Furthermore, α‐HH morphology and size were controlled by seeding in the mixed solutions. The formation of regular large α‐HH crystals with a low‐specific surface area significantly weakened the surface adsorption of P in solutions and further increased the P leaching efficiency. More importantly, the large stumpy α‐HH was identified as high‐strength gypsum with high added value. This work would provide innovative guidance to efficiently remove impurities from gypsum and pioneer a cost‐effective approach for clean and high‐value utilization of industrial gypsum residues. [ABSTRACT FROM AUTHOR]

Published
2023
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